The internet (world wide web) is, to put it simple, lots of computers connected together to exchange information. To know where to send information to each computer has a unique address called IP number (internet protocol number), or just IP for short.
The IP consists of 4 numbers separated by 3 dots: xxx.xxx.xxx.xxx, for example 217.162.125.141.
The company responsible for providing you with an internet connection (DSL, cable, ISDN) and an IP address is your ISP (internet service provider). Most of the time they will give you a dynamic IP which is randomly taken from their pool of unused IPs. When disconnecting and reconnecting you will most probably have a new IP assigned to your computer.
There are also static IPs that never change. Those are mostly used for computers which need to be reachable 24/7 over the same address, such as servers.
A computer whose primary task involves sending data is called a server. This could be a web server for serving web pages, a mail server for receiving and sending emails, or even a game server that handles online playing between multiple players.
Computers connecting to servers in order to receive data are called clients. Usually there's one server providing data for many clients as shown in the picture on the right.
Common differences between clients and servers are:
Any computer can be a server, even the one you're using right now. However, because servers usually operate under mission critical conditions and need a fast upload, personal computers with standard internet connections and dynamic IPs are not very well suited for this task.